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MARKUS KLUTE: Welcome
back to 8.701.

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So in this lecture,
I'd like you to have

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a first connection
between particle physics

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and the Lagrangian formalism.

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In classical mechanics,
you have seen

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that you can write down
the Lagrangian using

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the kinetic and the potential
energy of a particle,

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and from that derive
equations of motions.

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In quantum field theory,
you can translate this idea

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and derive Lagrangian densities.

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It's beyond the
scope of this class

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to do all the mechanics of this.

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We'll visit this topic
later in the class

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when we introduce the Higgs
mechanism, for example.

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And we'll be a little
bit more systematic then.

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I'm introducing the topic now
because it allows you to answer

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one of the homework questions.

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So you can just
follow this lecture

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and then you should be able
to answer the first question

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of the second p-set.

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All right, so you just have
to trust me at this point

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that you can write the
Lagrangian for a Dirac field,

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or Lagrangian density for
a Dirac field this way.

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One exercise would be
to use this and show

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that from this Lagrangian you
can derive the Dirac equation

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for a spinor field.

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But that's not what
we're trying to do here.

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You're trying to see
what's the effect is

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of having this Lagrange density
being invariant or unchanged

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under global symmetry.

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So we are able to
rotate our spinor

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field with a global phase.

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And we will see
that the Lagrangian

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doesn't change and the
consequence of this,

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which is if we can.

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This is exercising
Noether's theory.

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There's an overarching
global symmetry.

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And out of the symmetry follows
the conserved property--

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in this case, the current.

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All right, so we can
express the symmetry

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with infinitesimal
phase transformation,

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as shown here, for our fields
and for our adjunct fields.

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For the field and
the derivatives,

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then, you just
have to do the math

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and we find those
expressions, which we can then

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put back into our Lagrangian.

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First of all, we
write the change

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of our Lagrangian in this way.

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And as we just have
seen the Lagrangian,

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its invariant under
this transformation,

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and therefore the
change is going to be 0.

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So then we use this information
at this in the equations.

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We find this very
complicated-looking set

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of equations.

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OK, so now we get this.

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And then we can
rewrite the terms.

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So this is already with
a vision of what we would

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like to actually find later.

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So if we now look at the
terms involving the derivative

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with respect to du
mu of our spinor,

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we can express this
equation as shown here.

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And with that, we find
the next equation.

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I only show this for the spinor
not for the adjunct spinor.

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This looks exactly the same.

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But you, however, find
in this part here,

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this looks like
Euler-Lagrange equation.

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And this part needs to be 0.

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So we only have to worry about
this part of the equation,

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and the same for
the adjunct field.

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So this then leaves
this equation here

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where we have i
epsilon, a derivative

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of this part of the equation.

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And something like this you have
seen before in our continuity

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equation, something like this.

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It's our continuity equation.

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We discussed this in one
of the last recitation

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session, which leads us
then to conserve currents.

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And let's go one step further.

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If we now identify this
part as our current,

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we can then use the
partial derivatives

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of our initial Lagrangian.

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Now we're just calculating
those terms here.

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And we find that our current,
our conserved current

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is given by the adjunct
spinor, gamma mu spinor.

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And so what we have just
seen-- and this is conserved,

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so the derivative is 0--

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have seen that we have
a Lagrangian density,

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we have a global symmetry.

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And out of that, we find that
the current is conserved.

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So this is all I wanted to show
here in the homework set now.

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We start from a
different Lagrangian.

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So this is our Lagrangian
for a massive spin half

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particle, which satisfies
the Dirac equation.

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In the homework, we are
looking at a scalar particle,

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a massive scalar particle.

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And the exercise, however,
is very much the same.